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To prove convergence, split the integral into $[1,2]$ and $[2,\infty)$. On $[1,2]$, the integrand is continuous, so the finite piece is well-defined.

Conditions: The integral is $\int_1^\infty \frac{x-2}{x^3+1}\,dx$.; The comparison function is $\frac{1}{x^2}$.; The inequality holds for $x \ge 2$.

To prove convergence, split the integral into $[1,2]$ and $[2,\infty)$. On $[1,2]$, the integrand is continuous, so the finite piece is well-defined.

Conditions: The integral is $\int_1^\infty \frac{x-2}{x^3+1}\,dx$.; The comparison function is $\frac{1}{x^2}$.; The inequality holds for $x \ge 2$.